AUGMENTING PHASE SPACE QUANTIZATION TO INTRODUCE ADDITIONAL PHYSICAL EFFECTS MATTHEW P. G. ROBBINS Bachelor of Science, University of Lethbridge, 2015 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Physics and Astronomy University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Matthew P. G. Robbins, 2017 AUGMENTING PHASE SPACE QUANTIZATION TO INTRODUCE ADDITIONAL PHYSICAL EFFECTS MATTHEW P. G. ROBBINS Date of Defense: July 25, 2017 Dr. Mark Walton Professor Ph.D. Supervisor Dr. Saurya Das Professor Ph.D. Committee Member Dr. Kent Peacock Professor Ph.D. Committee Member Dr. Hubert de Guise Professor Ph.D. External Examiner Lakehead University Thunder Bay, Ontario Dr. Kenneth Vos Associate Professor Ph.D. Chair, Thesis Examination Committee Abstract Quantum mechanics can be done using classical phase space functions and a star product. The state of the system is described by a quasi-probability distribution. A classical system can be quantized in phase space in different ways with different quasi-probability distributions and star products. A transition differential operator relates different phase space quantizations. The objective of this thesis is to introduce additional physical effects into the process of quantization by using the transition operator. As prototypical examples, we first look at the coarse-graining of the Wigner function and the damped simple harmonic oscillator. By generalizing the transition operator and star product to also be functions of the position and momentum, we show that additional physical features beyond damping and coarse-graining can be introduced into a quantum system, including the generalized uncertainty principle of quantum gravity phenomenology, driving forces, and decoherence. iii Acknowledgments Writing a graduate thesis is a difficult undertaking; it requires perseverance, dedication, and the ability to withstand periods of self-doubt. Completing this thesis feels gratifying for it demonstrates that over the two years of work, I managed to make an original contribution to the study of phase space quantum mechanics and quantization. However, it was also sometimes immensely frustrating when I pursued paths that turned out to be dead ends or pored over my work trying to find missing minus signs. To make it through a graduate program where a single question consumes most of your attention, there is a need for breaks, such as teaching undergraduate labs, bouldering, play- ing badminton, or even hanging out with friends by going to the movies or having a games night. There also must be ways to find some reason to laugh each and every day. Early in my Master’s program, my friends and I found out that every day celebrates something, such as cookies, hot air balloons, or Daleks. We began a tradition of finding out what was celebrated each day as well as the birthdays and anniversaries of scientists, inventors and new discoveries. We then combined what we found into a giant holiday, and this led to some rather interesting (absurd) days, including: Fat Squirrel Sled Race Day (February 2), Windmill Diffraction Day (May 10), Stupid Space Monkey Day (May 16), Subatomic Lunar Snack Day (July 21), and Give a Bear a Burger Day (November 16). For this, I would like to thank my penguin diagram on my whiteboard (yes, I’m thanking a Feynman diagram) and the penguin team (you know who you are). You guys tried to keep me sane and did an awful job of it! I want to thank my supervisor, Dr. Mark Walton, for giving me suggestions as I con- ducted my research and for reading through my thesis at least two or three times during iv ACKNOWLEDGMENTS the editing process. I do believe that my thesis is much better as a result of your many suggestions. Listening to the comments you made will help me as I pursue a PhD program and begin an academic career. I thank the other members of my supervisory committee (Dr. Saurya Das, Dr. Kent Peacock) for helping me through my graduate program and giving me assistance as I con- ducted the research. I also thank Dr. Hubert de Guise for his comments on my thesis and useful suggestions. Special thanks goes Kai Fender for answering any mathematical questions that I had. I would be remiss if I did not thank Alissa Kuhn and David Siminovitch for reading over my thesis. I cannot forget to thank Catherine Drenth, Dakota Duffy, and Chad Povey for their humour and conversations. Finally, I want to thank my parents for giving me support and encouragement as I pursued my Master’s degree. v Contents Contents vi List of Tables viii List of Figures ix 1 Introduction 1 2 Phase Space Quantization 7 2.1 Motivation . .7 2.2 Properties of the Operator Quantization Map . .8 2.3 Weyl Quantization . .9 2.4 Physical Implications of Different Quantizations . 11 2.5 Wigner Transform . 12 2.6 Moyal Product . 14 2.7 Wigner Functions . 18 2.8 Example: Simple Harmonic Oscillator . 22 2.9 Time-Dependence of the Wigner Function . 25 2.10 Transition Operators and Weight Functions . 27 3 Coarse-Graining 37 3.1 Motivation . 37 3.2 The Husimi Distribution . 38 3.3 A Generalization of the Husimi Distribution . 46 3.4 Smoothing in the n ! ¥ Limit of the Wigner Function . 47 4 Local Transition Operators 52 4.1 Motivation . 52 4.2 The Damped Harmonic Oscillator . 53 4.2.1 Quantization of Dissipative Systems . 53 4.2.2 Augmented quantization of the Simple Harmonic Oscillator . 55 4.2.3 Eigenvalue Spectrum . 58 4.3 Effects of the Complex Transition Operator, Tg ............... 58 4.4 Calculating the ~ ! 0 Limit with Star Products . 60 4.5 Transition Operators Involving Position and Momentum . 61 4.5.1 Motivation for Generalizing the Transition Operator . 61 4.5.2 Transition Operator for Damping . 62 4.5.3 Eigenvalue Spectrum . 67 vi CONTENTS 4.5.4 Relation of the Local Transition Operator to the Weight Function . 68 4.6 Star Product with Position and Momentum Dependence . 70 4.6.1 Derivation of the Star Product . 70 4.6.2 Properties of the Generalized Star Product . 74 4.6.3 Converting to Quantizations with Global and Local Transition Op- erators . 76 4.7 Star Product for the Damped Harmonic Oscillator . 78 4.8 Generalized Uncertainty Principle . 80 5 Time Dependent Transition Operators 88 5.1 Motivation . 88 5.2 Time-dependent Transition Operators and Star Products . 89 5.3 The Driven Harmonic Oscillator . 91 5.4 Environmental Decoherence . 94 5.4.1 Scattering Decoherence . 99 5.4.2 Long Wavelengh Limit . 99 5.4.3 Short Wavelength Limit . 104 6 Conclusion 106 Bibliography 114 A Derivation of Ordering Rules 124 A.1 Weyl Ordering . 124 A.2 Born-Jordan ordering . 125 B The Weierstrass transform 128 C Damping Transition Operator 130 C.1 Linear Damping . 130 C.2 Non-linear Damping . 131 vii List of Tables 2.1 Different quantizations and their corresponding weight functions [5]. 35 2.2 Mapping from Weyl quantization to another quantization in phase space. This map takes the form ei(qq+tp) ! Tei(qq+tp), where T is the transition operator [5]. 36 2.3 Properties of star products. Each of these quantizations obey [q; p]?T = i~, which is the phase space analogue of Heisenberg’s commutation relation. The bar over f ?T g signifies the complex conjugate. The transpose refers ! to ¶$¶, while the Hermitian conjugate is the complex conjugate of the transpose. 36 6.1 Quantizations discussed in this thesis. These have previously been studied in great detail (see, for example, [5]) . 110 6.2 Augmented quantizations discussed in this thesis. The Husimi distribu- tion is designed to coarse-grain the Wigner function [31]. Ref. [33] first proposed the damping augmented quantization discussed in Section 4.2.2. The remainder are original and are designed to incorporate specific physi- cal effects during quantization. Note that a, b, c, and d are functions of the position and momentum. 110 viii List of Figures 2.1 The simple harmonic oscillator Wigner function for the first four energy levels. Note that, for n > 0, the Wigner function has negative values. 25 2.2 There are many quantization maps. This figure illustrates one map relating Weyl operator quantization (AˆW ) to a different operator quantization (Aˆ) by way of two phase space quantizations (AW and A). 29 3.1 The simple harmonic oscillator Husimi distribution for the first four energy levels. Notice that the distribution is nonnegative, in contrast to Figure 2.1 . 45 3.2 The generalized Husimi distribution for the first four energy levels of the simple harmonic oscillator. We have set ~ = 1 and h = 0:5. Notice the distribution still has negative values even though smoothing was done. 47 3.3 The generalized Husimi distribution for the first four energy levels of the simple harmonic oscillator. We have set ~ = 1 and h = 2. Unlike Figure 3.2, the distribution is non-negative because h > ~.............. 48 3.4 The Wigner function for n = 50 (left) and n = 200 (right). The inset plot depicts n = 200 for 0 ≤ r ≤ 0:1........................ 49 3.5 Normalized convolutions of Wigner functions smoothed with the maximum (red) and minimum (green) inter-nodal distances. We see that larger n cor- responds to a decrease in width and increase in height. This seems to in- dicates that n ! ¥ implies that a delta function-like distribution will be found. 50 4.1 The transition operator Tg operating upon the simple harmonic oscillator Wigner function for the first energy level. We have set g = 0:2. Note that there is both a real and imaginary part to the distribution function.